Safe Subgame Resolving in Multiplayer Games

Poker and other games with hidden information are too large to solve in one pass. Systems that play them at a superhuman level instead compute a complete strategy offline, the *blueprint*, and then re-solve the part of the game actually reached during play, refining the blueprint in real time inside that *subgame*. For two players in a zero-sum game this second stage, subgame resolving, comes with a safety theorem: refinement never increases how much an adversary can gain by exploiting the strategy. With three or more players acting independently (uncorrelated profiles) no such guarantee exists. Multiplayer systems resolve heuristically, and resolving can increase a profile's vulnerability, the largest gain any single player obtains by deviating. We prove the first conditional safety theorem for uncorrelated multiplayer subgame resolving, a post-hoc certificate for any delivered update. The vulnerability of the resolved profile ${\sigma }^1$ exceeds that of the blueprint ${\sigma }^0$ by at most four terms. The first is the activation term $A_i^S$, the subgame value the blueprint promised but did not realize. The other three are the structural tax $\Theta \le 2n{\delta }_{\infty }$, the summed best-response margin deficits \sum_q M_q}, and the resolver's blueprint-relative subgame suboptimality ${\epsilon }_i$. The tax is an oscillation of the total utility, bounded by $2n$ times the distance ${\delta }_{\infty }$ to the class of constant-sum polymatrix (CSP) games, whose payoffs decompose into pairwise constant-sum interactions. Constructions attain each term, and both branches of the maximum in the sharper bound, with equality; a certified adversarial search over 162 instances finds no violation. Every rake-free poker variant is pointwise constant-sum, so the sharp structural tax vanishes there.